Circles — Class 10 Mathematics
"A circle is the simplest yet most profound shape — perfect symmetry around a single point."
1. About the Chapter
After studying circles in Class 9 (basic definitions, chord properties), Class 10 focuses on TANGENTS to circles — lines that touch a circle at exactly one point.
Why Important
Tangent properties are used in:
- Wheel and gear systems
- Optics (light rays at tangent angles)
- Engineering designs
- Astronomy (planetary orbits)
2. Recap — Circle Basics
Definitions
- Circle: set of all points equidistant from a fixed point (centre)
- Radius: distance from centre to circle
- Chord: line segment with both endpoints on the circle
- Diameter: longest chord, passes through centre = 2 × radius
- Arc: portion of the circumference
- Sector: pie-slice region
Position of a Line Relative to a Circle
A line can be:
- Non-intersecting (does not touch circle)
- Tangent (touches at exactly 1 point)
- Secant (intersects at 2 points)
3. Tangent to a Circle
Definition
A tangent is a line that touches a circle at EXACTLY ONE POINT.
Point of Contact
The single point where tangent touches the circle.
Common Examples
- Tyre touching the road
- Coin balanced on edge of table
- Sun's rays just grazing the horizon
4. Properties of Tangents (KEY THEOREMS)
Theorem 1: Tangent is Perpendicular to Radius at Point of Contact
The tangent at any point of a circle is perpendicular to the radius drawn to the point of contact.
If OP is radius to point P on circle, and AB is tangent at P, then OP ⊥ AB.
Proof (Outline)
Suppose tangent AB touches circle at P. Among all line segments from O to AB, OP is the shortest (since others go from O to points outside circle). Shortest distance from a point to a line is perpendicular. So OP ⊥ AB.
Theorem 2: Tangents from External Point are Equal
The lengths of two tangents drawn from an external point to a circle are equal.
If P is external, and PA, PB are tangents (touching circle at A, B), then PA = PB.
Proof (Outline)
- OA = OB (both radii)
- ∠OAP = ∠OBP = 90° (tangent ⊥ radius)
- OP common to both △OAP and △OBP
- △OAP ≅ △OBP (RHS criterion)
- Therefore PA = PB
Number of Tangents from a Point
- Inside circle: 0 tangents
- On circle: 1 tangent
- Outside circle: 2 tangents
5. Worked Examples
Example 1: Find Tangent Length
A point P is at a distance of 13 cm from the centre of a circle of radius 5 cm. Find the length of the tangent from P to the circle.
Setup:
- OP = 13 cm (distance from centre to external point)
- OQ = 5 cm (radius to tangent point Q)
- OQ ⊥ PQ (tangent perpendicular to radius)
Apply Pythagoras in △OPQ:
- PQ² = OP² − OQ² = 169 − 25 = 144
- PQ = 12 cm
Recognise the 5-12-13 triple.
Example 2: Equal Tangents
From external point P, two tangents PA and PB are drawn to a circle. If PA = 7 cm, find PB.
- By theorem: PA = PB
- PB = 7 cm
Example 3: Inscribed Quadrilateral
A circle is inscribed in a triangle ABC, touching sides at D, E, F. If AB = 10, BC = 11, AC = 13, find lengths of tangents.
Let tangents from each vertex = x, y, z.
- AD = AF = x
- BD = BE = y
- CE = CF = z
Then:
- AB = x + y = 10
- BC = y + z = 11
- AC = x + z = 13
Adding: 2(x+y+z) = 34 → x+y+z = 17
- z = 17 − 10 = 7
- x = 17 − 11 = 6
- y = 17 − 13 = 4
Example 4: Tangent Equation Style
Two tangents from external point to circle are 8 cm. Find the radius if the point is 10 cm from centre.
- Tangent length = 8
- Distance from centre = 10
- Radius² = 10² − 8² = 100 − 64 = 36
- Radius = 6 cm
6. Real-World Applications
Engineering
- Pulley systems use tangent properties
- Gear designs based on circles and tangents
- Belt drives between two pulleys
Optics
- Light rays reflecting off lenses use tangent geometry
- Spectacle lenses designed with tangents
Astronomy
- Planetary orbit tangents
- Eclipses involve tangent lines from Sun to Earth/Moon
Construction
- Road curves designed using tangent geometry
- Arches in architecture
Modern Tech
- GPS uses tangent properties for distance
- Robotics uses circular motion with tangents
7. Common Mistakes
-
Forgetting perpendicularity
- At point of contact, TANGENT IS PERPENDICULAR to radius.
-
Tangent passes through centre
- NO. Tangent is OUTSIDE circle (except at point of contact).
-
Tangents from interior point
- From point INSIDE circle: NO tangent possible.
-
Length confusion
- Tangent length is measured FROM external point TO point of contact.
-
Pythagoras error
- In △OPQ: OP is hypotenuse (distance to external point), OQ is leg (radius), PQ is leg (tangent length).
8. Indian Context
Circles in Indian Mathematics
- Aryabhata approximated π to 3.1416
- Madhava of Kerala (14th c.) gave infinite series for π
- Indian Vedic geometry used circles extensively
Modern Use
- Indian Railways tracks use circular curves
- Highway design uses tangent transitions
9. Worked Example with Two Tangents
Example: Tangents from External Point
From external point P, two tangents are drawn touching circle at A and B. If angle APB = 60°, find angle AOB.
In quadrilateral OAPB:
- ∠OAP = 90° (tangent ⊥ radius)
- ∠OBP = 90° (tangent ⊥ radius)
- ∠APB = 60° (given)
- Sum of angles in quadrilateral = 360°
- ∠AOB = 360° − 90° − 90° − 60° = 120°
10. Conclusion
The geometry of tangents is elegant and PROFOUND:
- Tangent ⊥ Radius at point of contact
- External tangents are EQUAL in length
- Used everywhere — engineering, optics, design
Master:
- Two key theorems
- Pythagoras in tangent problems
- Equal tangent lengths
This chapter is high-yield for board exams. Practice 15+ problems.
Circles and tangents: geometry's most graceful relationship.
